Lec.14.pptx HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS

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14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS

I Main Topics

A PosiBon, displacement, and differences in posiBon of two points

B Chain rule for a funcBon of mulBple variables C Homogenous deformaBon D Examples

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x

y

U1 U2

X1(?)

X1’ X2’

X2(?)


II PosiBon, displacement, and differences in posiBon of two points A IniBal posiBon vectors: Pt. 1: Pt. 2:

B Final posiBon vectors: Pt. 1’: Pt. 2’:

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PosiBon Vectors

Point 1 moves to Point 1’ Point 2 moves to Point 2’

X1 =

x1 +y1

X2 =x2 +y2

X ′1 = x ′1 +

y ′1X ′2 = x ′2 + y ′2

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II PosiBon, displacement, and differences in posiBon of two points (cont.) C Displacement vectors

1 In terms of posi5ons: Pt.1:

Pt.2: 2 In terms of

components: Pt.1:

Pt.2:

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Point 1 moves to Point 1’ Point 2 moves to Point 2’

U2 is displaced, rotated, and “stretched” relaBve to U1.

Displacement Vectors

U1 =

X ′1 −

X1

U2 =X ′2 −

X2

U1 =

u1 +v1

U2 =u2 +v2


II PosiBon, displacement, and differences in posiBon of two points D Difference in posiBons

of Points 1 and 2 1 Difference in iniBal

posiBons dX = X2 − X1

2 Difference in final posiBons dX′=X2’ − X1’

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Point 1 moves to Point 1’, not Point 2 Point 2 moves to Point 2’

dX’ is displaced, rotated, and stretched relaBve to dX.

Difference in Posi5ons (not displacement vectors)

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II PosiBon, displacement, and differences in posiBon of two points

E Displacement gradient terms (in a matrix)

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Displacement Gradient Components

∂u∂x

∂u∂y

∂v∂x

∂v∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Gradient terms combine changes in displacement components with

changes in posiBon components

These describe how the components of U change

as the components of X change

14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION

GRADIENTS III Chain rule

A FuncBons of two variables

The total change in a funcBon of variables x and y equals its rate of change with respect to x, mulBplied by the change in x, plus its rate of change with respect to y, mulBplied by the change in y.

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z = z x, y( ); dz = ∂z∂xdx + ∂z

∂ydy

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III Chain rule

A FuncBons of two variables (cont.)

x’ = x’(x,y), y’ = y’(x,y) u = u(x,y), v = v(x,y)

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d ′x =∂ ′x∂x

dx + ∂ ′x∂y

dy

d ′y =∂ ′y∂x

dx + ∂ ′y∂y

dy

du = ∂u∂xdx + ∂u

∂ydy

dv = ∂v∂xdx + ∂v

∂ydy


III Chain rule

A Two variables (cont.)

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d ′x =∂ ′x∂x

dx + ∂ ′x∂y

dy

d ′y =∂ ′y∂x

dx + ∂ ′y∂y

dy


∂ydy


∂ydy

d ′xd ′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

∂ ′x∂x

∂ ′x∂y

∂ ′y∂x

∂ ′y∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

dxdy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

d ′X[ ] = F[ ] dX[ ]

dudv

⎡

⎣⎢

⎤

⎦⎥ =

∂u∂x

∂u∂y

∂v∂x

∂v∂y

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

dxdy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

dU[ ] = Ju[ ] dX[ ]

Matrix form

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III Chain rule

B Three variables

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d ′xd ′yd ′z

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

∂ ′x∂x

∂ ′x∂y

∂ ′x∂z

∂ ′y∂x

∂ ′y∂y

∂ ′y∂z

∂ ′z∂x

∂ ′z∂y

∂ ′z∂z

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

dxdydz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

d ′X[ ] = F[ ] dX[ ] dU[ ] = Ju[ ] dX[ ]

dudvdw

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

∂u∂x

∂u∂y

∂u∂z

∂v∂x

∂v∂y

∂v∂z

∂w∂x

∂w∂y

∂w∂z

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

dxdydz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Final posiBon in terms of iniBal posiBon Displacement in terms of iniBal posiBon


IV Homogeneous strain

A EquaBons of homogeneous 2-‐D strain 1 Chain rule

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d ′x =∂ ′x∂x

dx + ∂ ′x∂y

dy

d ′y =∂ ′y∂x

dx + ∂ ′y∂y

dy


∂ydy


∂ydy

1 Scale a At a point, derivaBves have unique

(constant) values; equaBons are linear in dx and dy in the neighborhood of the point (e.g., dx’ and dy’ depend on dx and dy raised to the first power.

b If the derivaBves do not vary with x or y, (i.e., are constant), then the equaBons are linear in dx and dy no mager how large dx and dy are. This is the condi5on of homogenous strain.

c Homogeneous strain applies at a point

d Homogeneous strain is applied to “small” regions

e DeformaBon in large regions is invariably inhomogeneous (derivaBves vary spaBally)

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A EquaBons of homogeneous 2-‐D strain (cont.) 2  Common reformulaBon

a For constant derivaBves a, b, c, d, replace dx, dy, dx’ and dy’ by x, y, x’, and y’ (derivaBves are the same for small dx or large x)

b Linearity is clarified c Chain rule origin is

obscured

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d ′x =∂ ′x∂x

dx + ∂ ′x∂y

dy⇒

′x = ax + by

d ′y =∂ ′y∂x

dx + ∂ ′y∂y

dy⇒

′y = cx + dy


A EquaBons of homogeneous 2-‐D strain (cont.)

1 Lagrangian a x’ = ax + by b y’ = cx + dy

2 Eulerian (see derivaBon)

a x = Ax’ + By’ b y = Cx’ + Dy’

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Final posiBons

IniBal posiBons

Final posiBons

IniBal posiBons

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DerivaBon of Eulerian equaBons a x’ = ax + by y = (x’–ax)/b b y’ = cx + by y = (y’–cx)/d Equate right sides above d (x’ – ax)/b = (y’ – cx)/d e d(x’–ax) = b(y’– cx) f cbx – adx = by’-‐dx’ g x(cb-‐ad) = by’-‐dx’ h x = [-‐d/(cb-‐ad)] x’

+ [b/(cb-‐ad)] y’ i x = Ax’ + By’

j x’ = ax + by x = (x’–by)/a k y’ = cx + by ! !x = (y’–dy)/c Equate right sides above l (y’ – dy)/c = (x’ – by)/a m a(y’ – dy) = c(x’ – by) n cby – ady = cx’-‐ay’ o y(cb-‐ad) = cx’-‐ay’ p y = [c/(cb-‐ad)]x’ + [-‐a/(cb-‐ad)]y’

q y = Cx’ + Dy’

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IV Homogenous (uniform) deformaBon (cont.) B Straight parallel lines

remain straight and parallel (see appendix)

C Parallelograms deform into parallelograms in 2-‐D;

D Parallelepipeds deform into parallelepipeds in 3-‐D

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IV Homogenous (uniform) deformaBon (cont.) E Circles deform into ellipses in

2-‐D (see appendix); Spheres deform into ellipsoids in 3-‐D

F The shape, orientaBon, and rotaBon of the strain ellipse or ellipsoid describe homogeneous deformaBon.

G The rotaBon is the angle between the axes of the strain ellipse and their counterparts before any deformaBon occurred (to be elaborated upon later).

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IV Homogenous (uniform) deformaBon (cont.) H Lagrangian equaBons

for posiBon 1

2

3

I Lagrangian equaBons for displacement 1

2

3

4

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′x = ax + by′y = cx + dy

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= a b

c d⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

′X[ ] = F[ ] X[ ]

F = DeformaBon gradient matrix

u = ′x − x = a −1( )x + byv = ′y − y = cx + d −1( )yuv

⎡

⎣⎢

⎤

⎦⎥ =

a −1 bc d −1

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=e fg h

⎡

⎣⎢⎢

⎤

⎦⎥⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

U = Ju[ ] X[ ]

Ju[ ] = F[ ]− I[ ], where I = 1 00 1

⎡

⎣⎢

⎤

⎦⎥

Ju= Jacobian matrix for displacements

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PosiBon transformaBons (Lagrangian)

Displacement equaBons (Lagrangian)

PosiBon transformaBons (matrix form)

Displacement equaBons

(matrix form)

DeformaBon gradient tensor F

Displacement gradient tensor Ju

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V Examples A No deformaBon

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1 0

0 1⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

uv

⎡

⎣⎢

⎤

⎦⎥ =

0 00 0

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 00 1

⎡

⎣⎢

⎤

⎦⎥

0 00 0

⎡

⎣⎢

⎤

⎦⎥

′x = 1x + 0y′y = 0x +1y

u = 0x + 0yv = 0x + 0y






(matrix form)



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V Examples B Rigid body translaBon

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1 0

0 1⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

cxcy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

uv

⎡

⎣⎢

⎤

⎦⎥ =

0 00 0

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 00 1

⎡

⎣⎢

⎤

⎦⎥

0 00 0

⎡

⎣⎢

⎤

⎦⎥

′x = 1x + 0y + cx′y = 0x +1y + cy

u = 0x + 0yv = 0x + 0y

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14. HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS Example C: Rigid body rotaBon




Displacement equaBons (matrix form)



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V Examples C Rigid body rotaBon

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= cos60° − sin60°

sin60° cos60°⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

cos60° − sin60°sin60° cos60°

⎡

⎣⎢

⎤

⎦⎥

cos60° −1 − sin60°sin60° cos60° −1

⎡

⎣⎢

⎤

⎦⎥

′x = cos60°( )x − sin60°( )y′y = sin60°( )x + cos60°( )y

u = cos60° −1( )x − sin60°( )yv = sin60°( )x + cos60° −1( )y

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= cos60° −1 − sin60°

sin60° cos60° −1⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥





Displacement equaBons (matrix

form)



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V Examples D Uniaxial shortening

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1 0

0 0.5⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

uv

⎡

⎣⎢

⎤

⎦⎥ =

0 00 −0.5

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 00 0.5

⎡

⎣⎢

⎤

⎦⎥

0 00 −0.5

⎡

⎣⎢

⎤

⎦⎥

′x = 1x + 0y′y = 0x + 0.5y

u = 0x + 0yv = 0x − 0.5y

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Example E: DilaBon





(matrix form)



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V Examples E DilaBon

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 2 0

0 2⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

uv

⎡

⎣⎢

⎤

⎦⎥ =

1 00 1

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

2 00 2

⎡

⎣⎢

⎤

⎦⎥

1 00 1

⎡

⎣⎢

⎤

⎦⎥

′x = 2x + 0y′y = 0x + 2y

u = 1x + 0yv = 0x +1y





Displacement equaBons (matrix

form)



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V Examples F Pure shear strain

(biaxial strain, no dilaBon

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 2 0

0 0.5⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

uv

⎡

⎣⎢

⎤

⎦⎥ =

1 00 −0.5

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

2 00 0.5

⎡

⎣⎢

⎤

⎦⎥

1 00 −0.5

⎡

⎣⎢

⎤

⎦⎥

′x = 2x + 0y′y = 0x + 0.5y

u = 1x + 0yv = 0x − 0.5y

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(matrix form)



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V Examples G Simple shear strain

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 1 2

0 1⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

uv

⎡

⎣⎢

⎤

⎦⎥ =

0 20 0

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1 20 1

⎡

⎣⎢

⎤

⎦⎥

0 20 0

⎡

⎣⎢

⎤

⎦⎥

′x = 1x + 2y′y = 0x +1y

u = 0x + 2yv = 0x + 0y






(matrix form)



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V Examples H General deformaBon (plane strain)

′x′y

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 2 1

0 −0.5⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

uv

⎡

⎣⎢

⎤

⎦⎥ =

1 10 −0.5

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

2 10 −0.5

⎡

⎣⎢

⎤

⎦⎥

1 10 −0.5

⎡

⎣⎢

⎤

⎦⎥

′x = 2x +1y′y = 0x + 0.5y

u = 1x +1yv = 0x − 0.5y

Lec.14.pptx HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS

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Transcript of Lec.14.pptx HOMOGENEOUS FINITE STRAIN: DISPLACEMENT & DEFORMATION GRADIENTS